## Saturday, December 4, 2021

### HP 17B and HP 27S: Derivatives to the Nth Order (and TI-84 Plus CE Python 5.7 update)

HP 17B and HP 27S: Derivatives to the Nth Order

With the Solver of the HP 17B family, HP 27S, and the HP 19B calculator, we can calculate derivatives of any order.  Four derivatives are presented here.  You can use any name you want other than those presented here.  Each derivative is to the kth order.

Derivative 1:   d^k/dx^k a × x^n

N and K must be positive integers, D is the value of the derivative

DER1: D=A×X^(N-K)×PERM(N:K)

Example:

Input:  N = 2, K = 1, A = 3, X = 1.5

Result:  D = 9

Derivative 2:   d^k/dx^k e^(a × x)

K must be a positive integer, D is the value of the derivative

DER2: D=A^K×EXP(A×X)

Example:

Input:  A = 1.8, K =3, X = 3

Result:  D = 213.4409

Derivatives 3 and 4 will require trigonometric functions, which are not available on the HP 17B family.   It is recommended you set the calculator to Radian angle mode.

Derivative 3:  d^k/dx^k sin(a × x)

DER3: D=IF(MOD(K:2)=0:(-1)^(K÷2)×A^K×SIN(A×X):(-1)^((K+3)÷2)×A^K×COS(A×X))

Examples:

Input:  A = 0.75, X = 0.66, K = 2

Result: D = -0.2672

Input:  A = 0.75, X = 0.66, K = 3

Result: D = -0.3712

Derivative 4:  d^k/dx^k cos(a × x)

DER4: D=IF(MOD(K:2)=0:(-1)^(K÷2)×A^K×COS(A×X):(-1)^(K÷2+1÷2)×A^K×SIN(A×X))

Examples:

Input:  A = 0.75, X = 0.66, K = 2

Result: D = -0.4950

Input:  A = 0.75, X = 0.66, K = 3

Result: D = 0.2004

Eddie

All original content copyright, © 2011-2021.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.

## Tuesday, November 30, 2021

### TI-84 Plus CE and CE Python 5.7 update

TI-84 Plus CE and CE Python 5.7 update

Texas Instruments has released software updates, 5.7, for both the TI-84 Plus CE and TI-84 Plus CE Python.

Updates include (not an all-inclusive list):

*  Friendlier treatment of entering negative numbers.  Example:  entering [(-)] 5 [(-)] 7 will return -5*-7 (and 35 as the answer) instead of an error.  In Mathprint mode, a multiplication symbol (*) is added to the input automatically.

*  Update to the CelSheet app.   I have not used the CelSheet app, yet, one more thing to add to the list of things I need to learn about.

*  For the Python edition, the ti_draw and ti_image modules (custom to Texas Instruments, both covering drawing and images) are added.  You can add the menus of the commands by going in Edit mode, pressing [ y= ] (Fns...), arrow to Modul, press [ trace ] (Add-On) and selecting the ti_draw and/or the ti_image module.   The calculator not only adds the menu commands to the Fns... menu but adds the required from ti_draw/ti_image import * commands.

Remember that Python files are saved in RAM, but can be achieved.

More details:

Tiplanet's page (in French):

TI-84 Plus CE Python 5.7 - you want to download the Apps bundle:

TI-84 Plus CE 5.7 - you want to download the Apps bundle:

Caution:  As with version 5.6, 5.7 does not include any support for ASM (assembly) programs.   Also, once updated, the calculator can not be downgraded.

All original content copyright, © 2011-2021.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.

## Sunday, November 28, 2021

### HP Prime: Graphs of Permutation and Combination

HP Prime:  Graphs of Permutation and Combination

Definitions

Permutation:

perm(n, x) = n! ÷ (n - x)!

Combination:

comb(n, x) = n! ÷ (x! × (n - x)!)

Combination with Repeated Choices allowed:

comb(n + x - 1, x) = (n + x - 1)! ÷ (x! × (n - 1)!)

For the following graphs, I use screen shots with the HP Prime Virtual Calculator.

Discrete Graphs

Let n and x be positive integers.

Continuous Graphs

N is still a positive integer while x is a positive real number.   The factorial can be represent as the Gamma function as  x! = Γ(x+1).

3D-Graphs

The variables  x and y take real values.

Eddie

All original content copyright, © 2011-2021.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.

## Saturday, November 27, 2021

### Casio fx-CG50: Integer Digit Operations

Casio fx-CG50: Integer Digit Operations

Sum of the Digits and Digital Sum with Memory

The sum of the digits is pretty much a straight forward calculation, separate all the digits and add them up.  Example:

N = 18,283

Sum of the digits: S = 1 + 8 + 2 + 8 + 3 = 22

Ernesto Estrada, author of the article "Integer-Digit Functions: An Example of Math-Art Integration" (see the Source Section below), introduces the sum of the digits with memory, which is defined as:

F = N × S

Going with our example, N = 18,283.  With S = 22,

F = 18,283 × 22 = 402,226

The program INTDGT, programmed on the Casio fx-CG50, calculates the sum of a digits of a positive integer and the digital sum of the memory.

Variables:

N = input

S = sum of the digits

F = digital sum with memory

List 1 = list where each element is a digit of N

Casio fx-CG50 Program:  INTDGT

"2021-09-21 EWS"

"N>0, INTEGER "? → N

N → A

Int log N → L

Seq(0,x,0,L,1) → List 1

For 1 → K To L+1

Int (A÷10^L) → List 1[K]

A-List 1[K]×10^L → A

L-1 → L

Next

Sum List 1→ S

S×N → F

"SUM OF DIGITS="

S ◢

"S×N="

F ◢

"DIGITS IN List 1"

List 1

Sum of Digits as Functions Applied to Each Digit

The program FNDGT calculates the sum of each digit and returns a list in a four element list:

{ Σ (n_k),  Σ (n_k)^2,  Σ sin(n_k), Σ cos(n_k) }

Radians angle mode is used.  The results are stored in List 2.

Casio fx-CG50 Program:  FNDGT

"2021-09-21 EWS"

"N>0, INTEGER"? → N

N → A

Int log N → L

{0,0,0,0} → List 2

For L → K To 0 Step -1

Int (A÷10^K) → M

List 2 + M → List 2

List 2 + M^2 → List 2

List 2 + sin M → List 2

List 2 + cos M → List 2

A - M×10^K → A

Next

"K, K^2, sin K, cos K" ◢

List 2

Example:

N = 93,259

Results:

{ 28, 200, 0.9157301307, -2.944737671 }

Integers and Art

I encourage everyone to read the article listed in the source section below.   Ernesto Estrada demonstrates how to turn the sum of a digits and applying functions to the digits into beautiful art:   grass, fish, asteroids, butterflies, and other beautiful patterns.  If you have the chance, get this article while it is Open Access!

Source:

Estrada, Ernesto  "Integer-Digit Functions:  An Example of Math-Art Integration"  The Authors, Volume 40, Number 1, 2018.   https://link.springer.com/article/10.1007%2Fs00283-017-9726-x

Retrieved September 15, 2021.   Open Access*

*as of 9/15/2021

In a few days the 2021 calendar will just be one page.

Eddie

All original content copyright, © 2011-2021.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.

## Thursday, November 25, 2021

### HP 17BII+: Update to Normal CDF Formula - Don Phillips

HP 17BII+:  Update to Normal CDF Formula - Don Phillips

On March 31, 2019, I posted a formula calculate the normal cumulative distributive function (normal CDF) for x ≥ 0:

Don Phillips provided us with an updated formula to include negative values of x.   Much gratitude for allowing me to post this formula:

NCDF=L(ANS:1-EXP(-ABS(X)^2÷2)÷SQRT(2×PI)×(.4361836×L(T:INV(1+.33267×ABS(X)))-.1201676×G(T)^2+.9372980×G(T)^3))×0+IF(X<0:1-G(ANS):G(ANS))

Example Calculations:

X = -2.50,  NCDF = 0.00622

X = -1.50, NCDF = 0.06680

X = -0.50, NCDF = 0.30855

X = 0.00, NCDF = 0.50000

X = 0.50, NCDF = 0.69145

X = 1.00, NCDF = 0.84135

X = 2.00, NCDF = 0.97724

Thank you Don, Eddie.

All original content copyright, © 2011-2021.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.

## Monday, November 22, 2021

### Retro Review: Casio fx-55 Fraction Mate

Retro Review: Casio fx-55 Fraction Mate

Quick Facts

Model: fx-55 Fraction Mate

Company: Casio

Years: probably the 1990s or early 2000s, predecessor to the fx-55 Plus*

Memory Register:  1 independent memory

Battery:  Solar with battery back up (LR44)

Screen:  8 digits, 4 digit numerator, 4 digit denominator, function indicator (+, -, ×, ÷, ^), equals and parenthesis indicators

Logic:  Algebraic

Comes with hard case and reference card

In 2018, I reviewed the fx-55 Plus:  https://edspi31415.blogspot.com/2018/08/review-casio-fx-55-plus.html

Right now there are plenty of fx-55 (original keyboard) for sale on auction sites for eBay at really inexpensive prices, sometimes as low as about \$2.00.

Target Audiences

*  Elementary and middle school students

*  Anyone who wants a few more functions than a regular four-function calculator, but not a scientific calculator

*  You want a calculator with an emphasis on fractions

Keyboard

The fx-55 Fraction Mate does not have a shift key.   Some of the keys are:

[ + ], [ - ], [ × ], [ ÷ ]:  Arithmetic keys

[ ( ], [ ) ]:  Parenthesis keys

[ x^y ]:  Power key with a caret indicator on the screen

[ 1/x ]:  Reciprocal

[ RAN# ]:  3 digit random number between 0 and 1

[ FIX ]:  Fix setting.  For float setting, press [ FIX ] [ . ]

[ RND ]:  Round number to fixed setting internally

[ 10^x ]:  Antilog function

[ X ←→ Y ]:  Switch operators

[ X ←→ M]:  Exchange number in the display with number in memory

[ % ]: Percent key, see below

[ > ]: Backspace key

[ a ], [ b/c ]:  Fraction entry keys

[ F ←→ D ]:  Fraction/decimal conversion

[ a b/c ←→ d/c ]:  Improper/proper fraction conversion

[ SIMP ]:  Fraction simplification key

[ ÷R ]:  Integer division, with quotient and remainder.  If both numbers are positive, then remainder acts as a modulo operator

[ √ ]: square root

[ x^2 ]: square function

Integer Division

Pressing [ ÷R ] will return both the quotient and remainder of a division calculation.  If both numbers are positive, then the remainder is the modulo.

Example:  55 [ ÷R ] 12 returns 4 R 7.  Then 55 = 7 MOD 12.

Percent Key

To add % to n, press:

n [ × ] p [ % ] [ + ]

To subtract % from n, press:

n [ × ] p [ % ] [ - ]

Simplify Key

The [ SIMP ] key simplifies fractions when possible.   The factor used to simplify the fraction is briefly displayed.  All fractions on the fx-55 FRACTION MATE are simplified manually.

Example:   1450/2900 =

1450 [ b/c ] 2900 [ = ]

Display:  = 1450/2900  Simp

The Simp indicator means the fraction can be simplified.

[ SIMP ]

2 brifely displays before 725/1450 Simp

Repeated presses of [ SIMP ]:

5,  145/290 Simp

5,  29/58  Simp

29,  1/2

Order of Operations

The fx-55 follows the order of operations.  This is great for students in algebra.

Example:

3 [ + ] 9 [ × ] 11 [ = ] returns 102     (multiplication/division first, then addition/subtraction)

Verdict

I like the fx-55 Fraction Mate.  It has a lot of additional functions that beyond the traditional four-function calculator:  parenthesis, powers, reciprocal, fractions.  The display and the numbers are big.  I like how the arithmetic and power functions are displayed.   Worth the buy.

Casio Ledudu's page for fx-55, including the manual:  https://casio.ledudu.com/pockets.asp?type=1176&lg=eng

Eddie

All original content copyright, © 2011-2021.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.

## Sunday, November 21, 2021

### Casio fx-CG 50: Double the Investment, Rule of 72, and Rule of 69

Casio fx-CG 50: Double the Investment, Rule of 72, and Rule of 69

Introduction

Are you familiar with the Rule of 72?   In finance, the Rule of 72 refers to a simple formula to determine approximately how many years it takes an investment to double in value.

The Rule of 72 is an approximate formula, stated as:

n = 72 ÷ I

We can determine the actual time value of money formula (without periodic payment):

FV = PV × (1 + I ÷ 100)^n

where:

FV = future value

PV = present value

I = interest rate

n = number of years

Let PV = 1 and at double the investment, FV = 2.

2 = (1 + I ÷ 100)^n

Solving for n:

ln 2 = ln [ (1 + I ÷ 100)^n ]

ln 2 = n × ln [ (1 + I ÷ 100) ]

n = ln 2 ÷ ln [ (1 + I ÷ 100) ]

This is an exact calculation.

There is another approximation, called the Rule of 69, and the approximation is this:

n = 69 ÷ I + 0.35

In Summary:

Exact Calculation:  n = ln 2 ÷ ln [ (1 + I ÷ 100) ]

Rule of 72:  n = 72 ÷ I

Rule of 69:  n = 69 ÷ I + 0.35

Comparison

A spreadsheet, generated using a Casio fx-CG 50, is shown for rates from I = 2% to I = 20% in increments of 1%.

The Rule of 72 is the most accurate at I = 8%, with an error of 6.4 × 10^-3.  At interest rate I ≈ 7.846871453, the Rule of 72 equals the exact calculation.

When I < 7.846871453, the Rule of 72 over-estimates.  Other wise the Rule of 72 under-estimates.

For I = 2 to at least I = 20, the Rule of 69 under-estimates, but the difference shrinks as I increases.

Here is a graphical comparison for values of I = 5% to I = 15%.

Source

Thomsett, Michael C.  The Real Estate Investor's Pocket Calculator 2nd Ed. AMACOM:  New York.  2018.  ISBN 9780814438893 (paperback)

All original content copyright, © 2011-2021.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.

## Saturday, November 20, 2021

### Arcsine and Arccosine in terms of Arctangent

Arcsine and Arccosine in terms of Arctangent

Some Motivation

I recall reading about the Sinclair Scientific Programmable, a vintage scientific calculator that was introduced in 1975.  This calculator just had three trigonometric functions:  sine, cosine, and arctangent.  It is the aim of this blog to fill in the blanks.

In this blog entry, angles will have radian measure.

For tangent, it's pretty easy use of the trig identity:

tan x = sin x ÷ cos x

Determining Arcsine

Imagine the right triangle shown below:

Then:

sin Θ = x

Θ = arcsin x

By the Pythagorean Theorem:

t^2 + x^2 = 1

t^2 = 1 - x^2

t = √(1 - x^2)

tan Θ = x ÷ t

tan Θ = x ÷ √(1 - x^2)

Θ = arctan(x ÷ √(1 - x^2))

Then:

arcsin x = arctan(x ÷ √(1 - x^2))

Determining Arccosine

Most calculators for the arccos function have the range [0, π].  To accomplish this, we are going to use the identity

cos Θ = sin(π÷2 - Θ).

Let w = cos Θ

Then:

w = cos Θ

w = sin(π÷2 - Θ)

arcsin w = π÷2 - Θ

Θ = π÷2 - arcsin w

Θ = π÷2 - arctan(w ÷ √(1 - w^2))

For any angle w:

arccos w = π÷2 - arctan(w ÷ √(1 - w^2))

Eddie

All original content copyright, © 2011-2021.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.

## Monday, November 15, 2021

### Retro Review: Canon F-73P

Retro Review: Canon F-73P

Quick Facts

Model: F-73P
Company: Canon
Years: 1982
Memory Register:  1 independent memory, 6 other memory registers; 1-6
Battery:  2 x LR44 or 2 x A76 batteries
Screen:  LCD, 10 digits
Logic:  AOS (Algebraic)
Programming Memory:  45 steps between 2 program slots

I want to give thanks to J N Jewelry and Watch Repair Center in Covina, CA for not only getting the calculator open but also to replace the batteries.   You rock!

Two ON-OFF Switches?

The Canon F-73P has two ON-OFF switches:  one as a mechanical switch on the top left corner of the calculator, and the regular OFF and ON/C buttons.

The switch turns on and off the permanent memory, which the manual referred as to the power switch.   When the power switch is off, all the calculator's memory is cleared.

The ON/C and OFF turns the calculator's screen on and off.   Using the keys does not affect the calculator's memory.

Features

The Canon F-73P calculator is a scientific calculator with the following modes:

*  Hyperbolic functions
*  Base Conversions:  Decimal, Hexadecimal, and Octal  (no binary)
*  Single Variable Statistics
*  Linear Regression Statistics (y = a ∙ x + b)
*  Programming
*  Metric/US Conversions

RV Function: The RV function ( [ INV ] [ M+ ] ) reverse the operands in arithmetic operations.  The manual says it is for multiplication and division, but it works for addition, subtraction, power (a^x), and roots (x√) as well.

Conversions:  A button  [ → ] ( ← ) is dedicated to four sets of U.S.-metric conversions.  Note that this the bold right arrow key, located above the [ STO ] key on the right side of the calculator.

[ + ]:   gallons < - > liters
[ - ] :   Temperature: °F < - > °C
[ × ]:  lb < - > kg
[ ÷ ]:  in < - > cm

Even though the calculator can have a bigger set of conversions, understandably, it would clutter the keyboard.   Grateful to have these conversions built in.

Programming

The F-73P has two programming slots that can contain up to 45 steps.  Unfortunately, the programming is "blind", with out a way to view or edit programs.   This means that extra care, planning, and testing may be required.

Entering programming mode, Mode 0 for Learn Program 0 and Mode 1 for Learn Program 1 will clear all the previous steps

Like the Casio fx-3600P and Radio Shack EC-4004, there are the ENT and HLT commands:

ENT:  Entry command.   In programming mode, the ENT enters a prompt.  You are then required to enter a dummy value (not counted as a step) before continuing with entering commands.  In program mode, just pressing the [ → ] (hollow key on the top row) will enter the ENT command.

Entering a program is not only entering the commands but running a test case at the same time.

HLT:  Halt.  Stops program execution and shows the display.

What separates the F-73P from the simple programming calculators of the 1980's are the test and loop commands.  Instead of just sending the program back to start, the loop commands can send the point forwards or backwards up to 9 steps.

x > 0 n:   Tests to see if the number is greater than zero.  If true, then skip n (-9 to 9) steps.  Else, go on to the next step.

x ≤ M n:  Tests to see if the number is less than or equal to the value stored in the independent register M.   If true, then skip n steps.  Else, go to the next step.

GOTO n:  Skip n steps.

To help, the F-73P has partially merged steps.  The following count as one step:

INV + command
HYP + command
x > 0 + command
x ≤ M + command
Conversion key + command

All programs are run in Decimal Mode (Mode 1).  [ RUN ] starts program 1 while [ INV ] [ RUN ] starts program 2.

[ RUN ] is then used for answering prompts and continuing execution.

I wish that these programming calculators would allow for editing or at least viewing.

Example Program:

f(x) =  x^2 if x ≤ 0; else √x

1:  ENT   (dummy value, say 1)
2:  x > 0  2     (if x>0 skip 2 steps forward)
3:  x^2
4:  GOTO 2   (skip 1 step forward)
5:  INV √
6:  HLT

I have not figured out how to use the skip command steps for more involved algorithms such as the quadratic equations.

Verdict

The F-73P is a good calculator with your standard features.  The programming module is good for simple algorithms.  I like the eight conversions and base conversions.

Source

Toth, Viktor T.   "Canon F-73P"   R/S Programmable Calculators.  http://www.rskey.org/CMS/8-programmable-calculators/281-f73p  Accessed October 23, 2021.

All original content copyright, © 2011-2021.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.

## Sunday, November 14, 2021

### Casio fx-CG 50: Functions as Polar Graphs

Casio fx-CG 50:  Functions as Polar Graphs

Introduction

The function y = f(x) and equations in the form f(x,y) = g(x,y) can be transformed into its polar form by applying the transformations:

x = r cos Θ

y = r sin Θ

It can be a challenge getting the transformed equation in the form r = w(Θ), but it seems to work best where f(x,y) and g(x,y) are polynomials.

The following are graphs generated with a Casio fx-CG 50:

Function, y(x):   green with connected line

Polar Function, r(Θ):  blue with dots

Graphs

# 1:

y = x^2, r = sin Θ ÷ (cos Θ)^2 and r = 0

# 2:

y = x^3, r = ±√(sin Θ) ÷ (cos Θ)^(2/3) and r = 0

# 3:

y = 3x - 4, r = 4 ÷ (3 cos Θ - sin Θ)

General:  y = ax + b, r = -b ÷ (a cos Θ - sin Θ)

# 4:

y = 1/3 * (x - 4), r = -4 ÷ (3 sin Θ - cos Θ)

General:  ay = x + b, r = b ÷ (a sin Θ - cos Θ)

# 5:

y = ±√x, r = cos Θ ÷ (sin Θ)^2

# 6:

y = x^2 + x, r = tan Θ ÷ cos Θ - 1 ÷ cos Θ

# 7:

y = 1/x, r = √(1 ÷ (cos Θ sin Θ))

Eddie

All original content copyright, © 2011-2021.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.

## Saturday, November 13, 2021

### TI-92 Plus: Solving Problems Step by Step

TI-92 Plus:  Solving Problems Step by Step

This program should work with the TI-89 family (TI-89, TI-89 Titanium, Voyage 200).

Some CAS Tricks Used

With the help of the TI-89/92 Plus manual and adding restrictions with the with ( | ) command:

ln(x^y) | x>0 and y>0 returns x

expand(ln(x∙y)) | x>0 and y>0 returns ln(x)+ln(y)

sin^-1(sin(x)) | x≥-π/2 and x≤π/2 returns x

tan^-1(tan(x)) | x≥-π/2 and x≤π/2 returns

(mod(2∙x-π,2∙π)/2) - π/2

√(x^2) | x≥0 returns x

Also, the part command can be used to pick out parts of an equation or expression.  Using left and right can extract sides of an equation.

Example:

left(2∙x+5 = 3∙y) returns 2∙x+5

right(2∙x+5 = 3∙y) returns 3∙y

TI-92 Plus Program: steps

The equations offered in this program are:

a∙x+c = b

a∙x+b = c∙x+d

a∙x^2 + c = b

y = a∙b^x

√(a∙x+c) = b∙x+d

x^2+(2∙a)∙x = b

1/x+1/a = 1/b

sin(a∙x)∙cos(a∙x) = b

Click on the picture below for the code:

Take care,

Eddie

All original content copyright, © 2011-2021.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.

## Monday, November 8, 2021

### Derivatives with Surprisingly Imaginary Results

Derivatives with Surprisingly Imaginary Results

Here are three derivatives of functions where complex numbers are involved with further algebraic simplification.

d/dx √(a - x)^(1/2)

d/dx √(a- x)^(1/2)

= 1/2 ∙ (a - x)^(-1/2) ∙ -1

= -1/2 ∙ 1 ÷ (√(a - x))

Going a step further...

= -1/2 ∙ 1 ÷ (√(-1) ∙ √(x - a))

With √(-1) = i ,  1/i = -i

= i ÷ (2 ∙ √(x - a))

d/dx  arcsin(x + a)

d/dx arcsin(x + a)

= 1 ÷ √(1 - (x + a)^2)

= 1 ÷ √(1 - (x^2 + 2 ∙ a ∙ x + a^2))

= 1 ÷ √(-x^2 - 2 ∙ a ∙ x + 1 - a^2)

Factoring out -1 in the denominator:

= 1 ÷ √((-1) ∙ (x^2 + 2 ∙ a ∙ x - 1 + a^2))

= 1 ÷ (i ∙ √(x^2 + 2 ∙ a ∙ x - 1 + a^2))

= -i ÷ √(x^2 + 2 ∙ a ∙ x - 1 + a^2)

d/dx e^(√(a - x))

d/dx e^(√(a - x))

= e^(√(a - x)) ∙ d/dx √(a - x)

= -e^(√(a - x)) ÷ (2 ∙ √(a - x))

With:  √(a - x) = i ∙ √(x - a) and e^(i ∙ Θ) = cos Θ + i ∙ sin Θ

= -e^(i ∙ √(x - a)) ÷ (2 ∙ i ∙ √(x - a))

= -e^(i ∙ √(x - a)) ÷ (2 ∙ i ∙ √(x - a))

=  i ∙ e^(i ∙ √(x - a)) ÷ (2 ∙ √(x - a))

= (i ∙ (cos √(x - a) + i ∙ sin √(x - a)) ÷ (2 ∙ √(x - a))

= (-sin √(x - a) + i ∙cos √(x - a)) ÷ (2 ∙ √(x - a))

Eddie

All original content copyright, © 2011-2021.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.

## Sunday, November 7, 2021

### Solving Equations with Radicals and Extraneous Solutions

Solving Equations with Radicals and Extraneous Solutions

Introduction

For equations such as:

x^2 = a

Solving for x requires us to take the root of both sides with the solutions:

x = +a and x = -a.

However when have something in the form of:

√(f(x)) = a

we can take the square of both sides and solve for x.

f(x) = a^2

Sometimes our solutions of x may not work out.  This is where we run into the case of extraneous solutions.

What is an extraneous solution? This is a solution that may appear to be a valid solution by solving an equation through valid methods, but the catch is that this solution does not solve the equation.   Think of an extraneous solution as a "false solution".  We are going to see some examples in the next section.

Principal Square Root

When we refer to the square root of a number, most of the time we are referring to the principal square root.   That is, the principal square root only refers to the non-negative square root.

Example:  The principal square root of 256 is:

√256 = 16

Calculators use the principal square root for their square root function.

Example Problems where Extraneous Solutions May Exist

In all the examples, √ is used for the principal square root.  In fact, unless when specified, the √ symbol will always refer to the principal square root.   Other root operators, such as cube root and quartic root, will operating the same way.

Example 1:

x - 3 = √(3x - 9)

Start by squaring both sides:

x^2 - 6x + 9 = 3x - 9

Subtract (3x - 9) from both sides:

x^2 - 9x + 18 = 0

The left side can be factored into:

(x - 6)(x - 3) = 0

Giving potential solutions of:

x = 6 and x = 3.

Checking both solutions:

x = 6:   6 - 3 = 3;  √(6*3 - 9) = 3

x = 3:   3 - 3 = 0;  √(3*3 - 9) = 0

In this first case, both solutions, x = 3 and x = 6, are valid.

Example 2:

√(5x - 1) + 7 = 3

An attempt to solve for x:

√(5x - 1) = -4

5x - 1 = 16

5x = 17

x = 17/5

But checking for when x = 17/5:

√(5 * 17/5 - 1) + 7 = √(16) + 7 = 11 ≠ 3

In this case, x = 17/5 is an extraneous solution, and does not solve the equation.

Example 3:

x + √(4x + 1) = 5

An attempt to solve for x:

√(4x + 1) = 5 - x

4x + 1 = 25 - 10x + x^2

0 = 24 -14x + x^2

0 = (x - 2)(x - 12)

Possible solutions:  x = 2, x = 12

Checking:

x = 2:  2 + √(4*2 + 1) = 2 + √(9) = 5

x = 12:  12 + √(4*12 + 1) = 12 + √(49) = 19 ≠ 5

Hence only x = 2 is a true solution, as x = 12 is extraneous.

The bottom line: when solving equations involving radicals and roots, it is best to check your answers.  This is not a good practice in school, but in life.

Eddie

All original content copyright, © 2011-2021.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.

## Saturday, November 6, 2021

### TI-92 Plus and TI-Nspire CX II: List Filters

TI-92 Plus and TI-Nspire CX II:  List Filters

For this particular post, I also have a TI-Nspire CX II CAS document which can be downloaded here:  https://drive.google.com/file/d/1Rm5vIWdUMg5OEyXi4VecNfuIGdiGRtan/view?usp=sharing

The functions are public and are available in the catalog.  Examples and syntax are included.

The Filters

filters(list, criteria): filters out a list

sumfilt(list, criteria): sum of selected elements of a list

avgfilt(list, criteria): arithmetic average of elements of a list

prdlist(list, criteria): product of selected elements of a list

dimfilt(list, criteria): returns a count of list elements that fit a criteria

This is similar to Excel functions filter, sumifs, averagefits, and countifs; there is no productifs in Excel.

The criteria is a string, using x as a variable.  Examples:

"x<10": select elements less than 10

"x≥20": select elements greater than or equal to 20

"10<x and x<20": select elements between 10 and 20, not including 10 or 20

"10≤x and x≤20": select elements between 10 and 20, including 10 or 20

TI-92 Plus Functions: Filters

Note:  I am using the symbol [| for comment; in the program editor, select F2, option 9.

TI-92 Plus Function:  filters

filters(list1,cr)

Func

[| list, criteria string with x

[| EWS 2021-08-13

[| filter function

Local list2,d,i,t,x

{}→list2

dim(list1)→d

For i,1,d

expr(cr)|x=list1[i]→t

If t Then

augment(list2,{list1[i]})→list2

EndIf

EndFor

Return list2

EndFunc

TI-92 Plus Function: sumfilt

sumfilt(list1,cr)

Func

[| list, criteria string with x

[| EWS 2021-08-14

[| sum-if filter function

Local list2,d,i,t,x

{}→list2

dim(list1)→d

For i,1,d

expr(cr)|x=list1[i]→t

If t Then

augment(list2,{list1[i]})→list2

EndIf

EndFor

Return sum(list2)

EndFunc

TI-92 Plus Function: avgfilt

avgfilt(list1,cr)

Func

[| list, criteria string with x

[| EWS 2021-08-14

[| average-if filter function

Local list2,d,i,t,x

{}→list2

dim(list1)→d

For i,1,d

expr(cr)|x=list1[i]→t

If t Then

augment(list2,{list1[i]})→list2

EndIf

EndFor

Return sum(list2)/(dim(list2))

EndFunc

TI-92 Plus Function:  prdfilt

prdfilt(list1,cr)

Func

[| list, criteria string with x

[| EWS 2021-08-14

[| product-if filter function

Local list2,d,i,t,x

{}→list2

dim(list1)→d

For i,1,d

expr(cr)|x=list1[i]→t

If t Then

augment(list2,{list1[i]})→list2

EndIf

EndFor

Return product(list2)

EndFunc

TI-92 Plus Function:  dimfilt

dimflt(list1,cr)

Func

[| list, criteria string with x

[| EWS 2021-08-13

[| count-if filter function

Local list2,d,i,t,x

{}→list2

dim(list1)→d

For i,1,d

expr(cr)|x=list1[i]→t

If t Then

augment(list2,{list1[i]})→list2

EndIf

EndFor

Return dim(list2)

EndFunc

Examples:

list0 = {2,4,5,6,9,10,11,13,14,15,18}

filters(list0,"x≤10"):  {2,4,5,6,9}

filters(list0,"5≤x and x≤15"):  {5,6,9,10,11,13,14,15}

sumfilt(list0,"x≤10"):  36

sumfilt(list0,"5≤x and x≤15"):  83

avgfilt(list0,"x≤10"):  6

avgfilt(list0,"5≤x and x≤15"):  83/8

prdfilt(list0,"x≤10"):  21600

prdfilt(list0,"5≤x and x≤15"):  2700

dimfilt(list0,"x≤10"):  6

dimfilt(list0,"5≤x and x≤15"):  8

All original content copyright, © 2011-2021.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.

### HP 17B and HP 27S: Derivatives to the Nth Order (and TI-84 Plus CE Python 5.7 update)

HP 17B and HP 27S: Derivatives to the Nth Order With the Solver of the HP 17B family, HP 27S, and the HP 19B calculator, we can calculate d...